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Monetary policy in a game-theoretic framework
Sijben, J.J.
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1993
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~Monetary Policy in a
Game-Theoretic Framework
by
Jac J. Sijben
Reprinted from Jahrbucher fur
Nationalókonomie und Statistik,
Vol. 210, No. 314, 1992
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Reprint Series
CENTER FOR ECONOMEC RESEARCN Board
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Economic Research
Monetary Policy in a
Game-Theoretic Framework
by
Jac J. Sijben
Reprinted from Jahrbucher fur
National~konomie und Statistik,
Vol. 210, No. 314, 1992
Jah~b. ~. ,Vanonaltik. u. Stat. (G. Fische~ Veslyg, Stur[ga.t 1992) Bd. (Vol.) ~1013--:
Monetary Policy in a Game - Theoretic Framework
Spieltheoretische Ans~tze zur Geldpolitik
By Jac J. Sijben'), Tilburg1. Introduction
In the last decade the classical controversy about the pre-announcement of rules versus a discretionary monetary policy has been revitalized. This renaissance was linked with the rise and tht further development of the rational expectations theory or new-classical economics on the one side and with its implicacions for the effectiveness of macroeconomic stabilization policy on the ocher side (Sijben, 1980). [n this concext empirical research with regard to the ineffectiveness-postulate (Lucas, Sargenc and
Wallacej was also strongly emphasized (Barro, 1977).
- In the sevencies the debace wich regard co the inefEectiveness of economic scabilization policy did repel gradually the tradicional Keynesian-monetarisc controversy about the intluence of monetary and fiscal policy in the sixties. However at the same time a new policy-issue came about which put the debace on monetary-policy behavior in a game-theoretic framew-ork. This issue refers to the fact thac in the course of rime policy-makers will be tempted to change cheir original policy-stracegy wich regard to the intlation-unemploymenc trade-off. In che literacure this question is known as "the cime inconsistency of optimal plans" (Kydland and Prescott, 1977 and Taylor, 198~) and is closely related to che concepts oE credibiliry and repucacion of the central bank. ~1c the end of the seventies its aaencion was exciced when in several countries central banks were changing their poliey-behaviour, accompanied by a sharp reduction in economic activiry. [n ocher words it appeared thac in pracnce a disinflauon policy was very costly, requiring large, albeit temporary, increases in unemployment. (1979-1983 in the UK and the US). This developmenc resulted from che fact that during the seventies the public had losc its confidence in the ceneral bank with regard to the uphold of monetary stabiliry and workers and business had adjusced wages and prices in accordance wich their inflationary expectations. In this strong inflationary environmenc a change of the monetary-regime, a disinflaáon strategy, will confronc poliey makers wich a rise in unemploymenc and a slackening of economic activiry. The mearung of che "time-inconsisteney problem" with regard co che macro-economic scabilizacion policy is expressed by Taylor (1983, p. 123) as follows, "The possible time inconsistency of optimal policy is one of rhe most important policy issues chat has emerged Erom reseaich on rational txpectacions in macro-economics".
?3-1 J. J. 5[JBEN
This essav deals with a non-technical review of the current issues in the international literature in the fiefd of the game-theoretic approach m monetary policy. Especiall,v the importance of credibility and reputation of policy makers wirh regard to the formulanon and the implementation of monetary policy will be emphasized. According to this approach the interaaion process between the strategic behaviour of the policy maker and the public will result in an endogenous monetary poli~y. The tradicional view according to which the public responds to monetary impulses in a rather mechanical and predictable way will be abandoned.
Based on the stimulating articles of Kydland-Prescort on the one side and of Barro-Gordon and Backus-Driffill on the other side, in section Z the modern game-theoretic approach in monetary policy will be presenced. In rhis section it will be shown that in an environment with rational economic agenrs the choices of monetary authorities with regard to their policy behavior in a Phillips-curve analysis will be restricred essennally. This results from the fact thac a discrecionary monetary policv has to be abandoned, because in the long run the macroeconomic pay-off in terms of inflarion and emplovment will be less beneficial than in case of scicking to a pre-announced monetary growth rule. This conclusion is not based on Friedman's previous well-known arguments with regard to the variable lags and the timing and uncerrainry in monetary policy, but is rather based on the game-theoretic víew that "economic planning is not a game against nature but, rather, a game against rational economic agen[s". (Kydland and Prescott, 19", p. ~~3). Section 3 deals with the importance of asvmmetric information berween the policy maker and the public on the one hand and with the impact of the degree of mdependence of the central bank on the effectiveness of monetary polie~r on rhe other hand. Finally section -~ concludes with some summarizing remarks.
2. A Game-theor'etic Appsoach
Credibiliry
Since the introduaion of the racional expeaations theory (Lucas, Sargent, Wallace) and the concomitant Lucas-criuque in the seventies, much attencion has been given to the impact of credibiliry of policymakers in analysing the quesnon about rules versus discretion in monetary policy. The last few years the concepts of credibiliry and resoluteness of policy actions have been emphasized very strongly in considerations about the formulation and implementation of monetary policy. Fellner, who intro-duced this concept in macroeconomia, has pointed out thac the adjusement process during che transition period to lower inflation will be longer and more costly in terms of a loss of production and employment the less credible the anti-inflation policy ( Fellner, 1979 and McCallum, 1984). This concept is very crucial because che inflationary expectations of the rational public will determine ultimately the abiliry of policymak-ers, the central banks, to achieve their final objecrives. [n their turn these inIlationary expectations are dependent on the public's assessmenc of the credibiliry of cencral bank behavior.
~tonecary Policy in a Game 235
in which policy is acrually being conducted and to official announcements about its conduc[" ( McCallum, 1984, p. 105). This means that a new monecary policy-regime will be credible as it is believed immediately, notwithstanding the fact whether this new policy involves higher or lower inflation. Subsequently it is also very important that each moment the policy is consistent with the information of rhe public with regard to the objectives and limitations of the central bank. Therefore a monetary policy announcement will be less credible when the public is convinced that the policy intention is incompatible with the currcnc targets of policymakers ( Cukierman, 1986).
In the course of the seventies, characterized by a high inflationary environment in the industrializcd countries, central banks got increasingly convinced of the significance of maintaining the credibiliry of their anti-intlation policy. ln this decade the rate of intlation has been increased by the rise of oil-prices and excessive wage-demands on the one side and by an expansive fiscal policy on the other side. In these circumstances the public became very sceprical about the possibilines and in particular about the willingness of central banks to maintain monetary stabiliry. During the seventies two views have been put forward co reduce and to eliminate the accelerated rate of intlation. On the one hand an anti-intlation programme aiming at a gradual and predictable reduaion of the rare of intlation ( gradualism) and on the other hand a more radical programme emphasizing an immediate reduction of the rate of intlation ( commitment). These policy recommendations retlect the different views with regard to the speed of wage-and-price adjustments (contraca, institutions) and the concomitant costs in terms of produczion and employment. The success of both programmes is crucially dependent on the degree of policy-credibiliry. .~1 lack of credibiliry will hinder a sufficient reduction of inflationary expeaations, increasing the probabiliry of a recession. [n the seventies monetary policymakers in several counrries have conceded the wage- and price demands of trade unions and firms and the budget-deficits of the governments, feeding the arise and the self-generating inflationary processes. It is obvious that in such an inflationary environment the classical debate about rules versus discretion in monetary policy will be revitalized. The game-theoretic approach in monetary policy was initiated by the Lucas-critique and the concomitant interaaion becween the strategies of the players of the game (policymaker and the public).
The Kydland-Prescoct Analysis
As is well-known from the literature the issue of the most appropriate monetary policy strategy has always been strongly emphasized in economics. ~lfter the secónd world war, based on Keynesian cconomics, it was put forward that in the framework of economic stabilization policy a discrctionary policy could always replace a fixed monetary growth-rule. For it was reasoned ~hat smart policymakers could always make use of the lack of fuU information abouc the working of the economic process (private information) in implementing monetary policy. In this context Barro notes, "there is a substantive role for monetary policy only ro the extent thac the monetary authoriry has better information than the public" (Barro, 1976, p. Z6). A decade later he poincs ouc this vie~v as follows, "Then if the policymaker was also well-meaning, therc was no obvious defense for using a rule in order ro bind his hands in advance" (Barro, 1985, p-23).
236 J. J. SIJBEN
eighcies their analvsis had a strong influence as well on the theory of macro-economic stabilization policv as on the issue of internarional coordination of economic policy. The nucleus of the recent research on the game-theoretic approach in monetary policy refers to the costs of disintlation policy in terms of production and employment on the one hand and ro the question how to formulate and implemenr this policy co minimize chese costs on the other hand. Kydland and Prescott point out that a credible and steadv preannounced monetary policy rule always generates a better pay-off in terms of intlation and employment than a discretionary policy, when the macroeconomic outcomes are dependent on the expectations with regard to future monetary policy (Lucas-cririque). In this context the several political institutions also play a crucial role in formulating and implementing economic policy and in the ultimate course of the economic process. :~lesina and Tabellini explain it as follows, "The basic motivation of the novel literature on credibiliry and politics is that it is not harmless ro abstract from political institutions in modelling economic policy. The recent theorv of economic policy views policy making as a game becween the policymaker and the private agents in the economy" (Alesina and Tabellini, 1988, p. 452 and ,~lesina, 1987). In the last decade the game-theoretic concept of credibility also played a significant role in the contest of the need for an mternational coordination of economic policies to correa the financial imbalances in the world economy. This results Erom the fact thar a svstem of flexible exchange rates has appeared not to be able to eliminate the mutual interdependence of economic policies between countries. (~tiller and Wallace, 198~ and Hamada, 1974). Vloreover the game-theoretic approach is also practised in the interactions between monetary and fiscal poliey (central bank versus the Treasury) (Tabellini, 1986).
The following considerations are based on two elucidate essavs of Cukierman and Barro-Gordon respectively, which point ouc the essential characteristics oE rhe Kydland-Prescoa game-theoretic approach quite concisely (Cukierman, 1986, Barro-Gordon 1983-b and Backus-Driffill, 1985). In the basic-model macroeconomic policy is characterized as a non-repeated game (single stage or one-shot game), in which the cencral bank determines the actual rate of inflation (;t) and the public builds-up its indationary expectations (z'). The game consists of rhe players (the policymaker and the public), the strategies and the outcomes of the players. It is obvious that the time-dimension plays a critical role in game theory. In a non-repeated game the game is played only one time. This means that no confrontations betwcen the players have occurred nor in rhe past nor at presenc, so avoiding che influence of confrontacions on actual behavior. It also implies that in this simple game strategic interactions between the players will noc come about. Moreover this analysis refers to a non-cooperative game, implying that there is no precommitment about rules of the game. In this way che players are quite free to choose their optimal policy-strategy. However in a cooperative-game the possibiliry exists to tune the strategy-choices to each other.
The basic-model, focusing on the Phillipscurve trade-off between inflation and employment, consists of the following Eundamental relations.
YF - YN t (n - ;t') (1j
~ YF: actual produaion YN: natural production
Monerary Policy in a Game 237
This equarion refers to the well-known Lucas-supply function pointing out that deviauons from the natural level of produaion ( employment) are positively related co che unexpected race of intlacion. In other words expectations-errors with regard co intlation generace real etfecrs. This view is based either on a short-run Phillipscurve according to Lucas ( general versus relative prices) or on a supply-funaion in che context ot the contract cheory according to Fischer-Taylor with price and wage rigidiry~) (Sijben, 1980). For simpliciry ir is assumed chat the rate of monetary growth ( m) always equals the rate of intlacion ( :t), while the expected rare of monetary growch (m') derermines rhe expected rate of inflacion (tt').
Subsequently an objective function of the cencral bank is assumed according to which monetary auchoricies attach a negative weight co inflati~n while they value an increase of production and employment (YF 1 Y~) positívely. The macro-economic ouccome (Uo) of the policymaker can be described as follows:
Uo--zzta(YF-Y;~),a~0. (2)
The chird equarion refers to che objective function of the public, emphasizing a rational public which dislikes to be fooled by the policymakers by organizing a surprise intlation (a - Z' ~ 0). In other words che public prefers a zero-surprise intlauon which, according co equacion ( 1), implies an actual rate of unemploymenr equal to the natural rate of unemployment. The outcome of the public ( UP) can be rendered as follows:
UP--(Z-a')'- (3)
The public knows rhe poliey behavior of the cenrral bank and will accordingly decermine the inflationary expectations ( n`) which play a crucial role in the process of wage-demands. .~ zero intlation and the concomicant monecary policy will be considered optimal because the public dislikes intlation. [n these circumstances the aaual and the expected rate of intlauon will be zero ( rc -;t' - 0) and so there is no fooling of the public. This means char che level of aaual production coinddes with the level of natural produaion ( YF - YN) (equation 1).
Based on this model monetary policy will become endogenousl,v. This results from the Eact thar che polieymaker is maximizing its objective function caking inro account the rational behavior oE the public. This implies an inceraction becween the polieymaker and che public which, according to the game-theorenc analysis, can be descnbed in rhe following way (Backus and Driffill, 1985). As was pointed ouc before policymakers and the public are involved in a macroeconomic non-cooperacive game aimed at the fixation of a rate oE inflation and an employmenc level.
Substitutingequation (1) in (2) the foUowingobjective function of the policymaker in rerms of the rare of intlation results:
-U~- -z-fa(a-a'),a70.
r! Thc,Lucas-supply relarionship can also be formulaeed as a varianr of Friedman's narural ra[e
ot unemploymenr hyporhesis:
?38 J. J. SIJBEN
~lccording to the Backus-Driffill analysis ( 1985), the factor a represents the policv-preference or the weights polieymakers attach in their objective funezion to inflation and employment. It is assumed that a has a numerical value of 2. A higher numerical value of the parameter a represents a higher prioriry given to an elimination of unemployment and a stimulation of economic activiry above monetary stabiliry.
It is assumed that monetary authorities in formulating their policy know the public's expected rate of inf(ation. This means that the results of the decisions of policymakers are crucially dependent on the numerical values of the intlationary expectations of the public (a'). Owing to the fact that there exist two inElationary or monetary regimes (a - 0 and a- 1) and two values with regard to inflationary expectauons of the public (:t' - 0 and :t' - 1), each player of the game has four possible outcomes. These pay-offs will be deduced with the aid of a simple numerical example.
If the monetaty authorities fix a zero rate of intlation, implementing a policy of monetary stabiliry, then the following macroeconomic pay-offs will result (Table 1). If in these circumstances the public expects a zero race of inflation ( z' - 0) the final outcome will also be equal to zero. This monetary policy strategy prevents inflation and does not involve a real effect ( production). However if the public takes into account an expected rate of intlation equal to 1(:t' - 1), the policymaker's outcome will be equal to -? because of the working of rhe negative Lucas' supply-effect. although there is no intlanon, the real effecx will be negative with a contraaion m economic acnviry because of a negauve surpríse intladon. However if the policymaker chooses an intlationary policy ( z - 1), while the public expects a zero rare of inflation ( ;t' - 0), chen owmg ro the positive Lucas' supply-effect the policymaker's outcome will be equal to 1. [n these circumstances intlanon results but also a positive real effect m terms of an increase in producrion and employment. [f in this policy regime the expected rate of inflation is also equal to 1(:t' - 1) then the policymaker's outcome will be equal to -1. There is no real effect and only inflation results. The surprise-effect with regard to production and employment does not arise, but only the negative inflation effect will be obtained.
Based on thís simple numerical example it appears from the policymaker's view that an inflationary policy is more advantageous than a policy of monetary stabiliry, notwithstanding the inflationary expectations of the public. The macroeconomic outcome of the inflationary-strategy of the policymaker ( 1, -1) is always better than the zero rate inflationary-strategy (0, -2).
The ultimate results of the policymaker's strategies characzerized by alternative values of the chosen monetary regime and alternacive valua of the expecxed rate of inflation of the public can be summarized in the following matrix.
Table 1: Policymaker's Outcomes
~tonecary regime ( rz) Inflationary expeaaáons (n')
n'-0 a'-1 a - 0 ;s - 1 0 1 -2 -1
~lonerary Policy in a Game 239
This means thac the public does nor agree with a real wage rate that deviates from its eqwlibnum level, featured by an acutal level of producrion thac equals the natural level (YF - YH). In decermining their strategy it is assumed that economic agents know the actual rate of intlation (z). Based on the expcrience from the past the public knows the policymaker's prioriry of production and employment above monetary stabiliry (anti-inflation poliey). Therefore economic agents assume that the strategy of cheatíng with a concomitant surprise inflacion will always be followed (n - 1). For this inflanonary policy, as was mentioned before, gives the best outcome for the policymaker. This information leads to an expected positive rate of inflation by the public (:t' - 1). The ultimate result of this public behavior for the polieymaker's outcome is equal to -1 (see Table 1) and according to equation (3) equal to zero for che public (see Table 2).
The ultimate outcomes of the public, given the choice-strategies of the polieymaker, can be rendered as follows.
Table 2: Outcomes of thc Public
Vtonetary regime (a) Intlanonary expeaanons (a')
:r'-0 n'-1 z'~ z-1 0 -l -1 0
:ictu~l policy Expectacions of che public
zcro-intlation positive inflanon
zero-mFlation Parero-equilibrium Time-inconsistency
(pre-commitment) YF - YN YF ~ Y;~
z-n'-0 rzcn' posiuve intlation Time-inconsistency Time-consistency
(chearing) ( Nash-eqwGbnum)
YF~Uv YF-Yv
Jria' R-JI' 7O
z~o
1. J. s1Jae~v
which each plaver's straregy is opnmal given the other's strategy. The hierarchy of the players is equal. The ~Iash-equilibrium is the onlg sustatnable equilibnum that wtll arise when the policymaker has changed his origlnal optimal plan and thece is no room left for a surprise intlation'-).
The "Time Inconsistencv" Problem
In the foregoing sub-seaion it was put forward that a monetary policy which does not take into accounr the reaczions and expectations of the public with regard to future monetary policy will generate a worse result than a policy based on fixed rules. In this way the question of "time inconsisteney" in monetary poGcy came about, which can be explained as follows.
The temptanon of policymakers to cheat the public and to take advantage of a change in their strategy is always existent, notwithstanding the numerical values of the intlationary expectations of the public. However because of the fact that the public has learned from experience the incentive structure of the policvmaker with regard to production and employment, rational agents will not err systematically and will expect a positive rate of inflation (;t' ~ 0). In other words when the public has got to know the polictimaker's strategy, the authorities can not use rhe tactic of organizing a surprise intlation svstematically to stimulate economic activity. [t may be obvious that this game theorenc view links up with Friedman's fixed monetary growth rule as well as wtth the policy implications of the rational expectations theory. The behavior of the public, charaaerized by anticipating the policymaker's preference, will stimulate authorines to implement actually an inflationary policy to maintain production and employment on the natural level. Ultimately a vertical Phillipscurve results on the natural level of unemployment. For if in these circumstances policymakers do not accomodate the expectations of the public a negative surprise inflation (z e z') with a concomitant contraccion in economic activiry will occur which is considered as a negative macro-economic outcome (see Table 1).
This monetary policy strategy (inIlation-scenario) and the concomitant public behavior will generate the same level of employment (YF - Yh) as would be the case with a monetary growth rule (zero-inflation scenario). However the actual rate of inflation will be higher implying a poorer macroeconomic result. For as was pointed out in the previous section the outcome n- n' - 1(Nash-equilibrium) is inferior to the outcome a- a' - 0(Pareto-equilibrium). In this context Kydland and Prescott (1977, p. ~t74-475) remark, "... the resulting policy was consistent but suboptimaL It was consistent in the sense that at each point in time the policy selected was best, given the current situation", and further on, "Doing what is best, given the current situation, results in an excessive level of inflacion, but unemploymenc is no lower than it wotild be if inflation (possibly deElation or price stabiliry) were at the social optimal rate".
The dilemma described by Kydland and Prescott that the optimal monetary poliey strategy (zero-inflation) is time-inconsistent, while the time-consistent inflationary
~tonerary Policy m a Gamc ?4l
policy (discretion) is sub-oprimal has strongly stimulated further research in the field oE policy-credibiliry. The issue of credibiliry and the concomitant problem of "time-inconsisteney" can be formulaced as follows. Before the public has made its choice with regard to inflationary expeaations, a parcial optimal monetary policy strategy will elicit a reacrion from the public. This implies ex-post thar the reaaion on this initial optimal policy may be quite different from rhe reaaion the policymaker was expecting. In other words in this way the policymaker will be confronted wich quite another environment, because the public need not react in such a mechanical way as was assumed in the traditional macrceconomic models ( Lucas-cricique). This means that in this new sicuation policymakers will be stimulated co change their original optimal strategy and so renege on the initial plan, incroducing the poliey-credibilirv problem. [n this context Persson points ouc (1988, p. .i 20) "Presenc some imperfection which makes the ex-ante optimal policy a? nd besc rather than a 1 st best outcome, there is an ex-post incentive to deviations from the ex-ante policy", and further on "Therefore those policies that would be optimal if binding commirments could be made, face a credibilitv problem because of the incencive for ex-posc deviations. Forward-looking rational agencs only believe a policy announcement that will be optimal to carry ouc ex-post".
[n other words the temptation of policvmakers to drive the economv in the direction of the " first best" solution will ultimatelv drive rhe equilibrium from the "second best" co the " third best" solution because of the interaction process between the policvmaker and the public. In economic literature this development is known as the "cime-inconsisrencv" problem'). The concept of "time-inconsistencv" refers ro the rempta-rion of policymakers to deviate from their original plan when they rxpect that rauonal economic zgents will take into account this policy in their behavior. ~Iaintaining policy credibilicy in the sense of ex-post opnmality or time-consistent behavior of policyma-kers means an addirional constraint to the policv problems confroncing central banks. This implies that in each period policymakers have to determine rheir optimal strates,,ry based on the infiationary expectations and decision rules of rhe public. In this wav
policy will be endogenous. ~
Barro and Gordon point out that in the ultimate rational equilibrium situauon the monetary authorities can choose any desired rate of intlation. However if the cedibiliry of policymakers can be guaranteed beforehand, for example by way of the introduction of a monetary policy rule ( precommitment) binding the central banks in their policy behavior, the optimal solution ( zero-inflation scenario) can be achieved. For in case of an absence of such an institution or of issuing a credibiliry-guarantee by the central bank a positive rate of inflation and a sub-optimal solurion will result ( Nash- or discretionary equilibrium).
Theretore Kydland and Prescott emphasize the significance of fixed rules in monetary policy which signal credibiliry of policymakers. This preference for rules above discretion in monetary policy and its concomitant "inflationary bias" is essentially based on the fact that the economic system cannor be characterized as a"black-box" with á given dynamic structure. This means that the current decisions of economic
~) See, Cbnri, 1938, p. 18, where he notts, "... To re-evaluate the policymakers calculare the
?4'-
~. J. si1aEN
agents are conditioned by their future expectations. Therefore when monetary authorities in formulacing thcir optimal monetary policy ignore the racional behavior of the public, atterwards it will appear thac the original policy has co be changed.
It is obvious that a monetary policy behavior characterized by cheating the public is not cime-consistenc and will deteriorate policymaker's credibility and increase the uncertainty in the econo:ny. However a policy behavior which is cime-consistent will be credible and predictable giving a real contribution to an eliminacion of inflation. The message of Kydland and Prescott is pointed out by McCallum in the following way, "... Thus the surprise magnitude is zero on the avecage, over any large number of periods, even rhough the monetary authoriry views it as controllable in each period. The discretionary outcome, it is clear, features more inflation but the same amount of surprise inflacion (on the average) as under a rule. Accordingly to chis model, then, a discretionary mode of policy behavior by the monerary authority leads to consequences that are unambiguously poorer than would obtain (for the same economy and the same objectives) under rule-like behavior". (McCallum, 198~, p. 11~). The most important feature of a policy-rule refers to the fact that in this way future monetary policy behavior is clearly fixed and a discretionary policy wich its relatively sub-optimal macroeconomic outcome can be avoided. For assuming a fluctuating monetary regime, each period [he monetary authoricy will be confronted with the inflacion-unemploy-ment trade-off and will act accordingly. However because of the fact that the public knows the policvmaker's preference for production and employment above price stabilicy, a systematic surprise inflation cannot occur. Based on the informacion about poficy behavior the racional economic agents will adjust their intlationary expectanons to eliminare a consistent pattern of surprise inflation. ~ discretionary policy behavior featured by erratic changes in the monetary regime does noc offer the public a guarantee concerning che future behavior of monecary authorities. Therefore an inscitucional change (an independent central bank) or a monetary contract between che policymaker and the public with regard to polir behavior (price- or money supply-rule) will be necessary to prevent potential ex-post "surprises" and the concomitant discrecionary outcomes and to re-establish credibiliry. If such a restructuring of the institurions or rhe introduction of a formal rule is difficult or impossible to realize, then che question comes about whether it may be possible to find a more informal discipline-mechanism that can replace a precommitment or a binding contract. In these circumstances it might be possible that building up a reputation by policymakers and the concomitanc reputational forces can give a solution. For a loss of credibility, as will be described further on, will imply substantial macroeconomic costs and will bc punished by the public.
íVlonetary Policy in a Game 243
3. Asymmetsic In~ormation
Reputacional Forces
In the foregoing section it was assumed chac policymakers and the pub(ic have the disposal of the same informacion-set. In practice, however, this is mostly noc the case so that a sicuation of asymmecric informaàon exísts. This means that the public does noc know which kind of policymaker wiU be the opponent in the game and the policymaker in his curn does not know public's behavior with regard to inflacionary expectacions and che concomitant wage-demands. For a new monetary authoriry may have quite anocher pceference with regard co che macroeconomic crade-off between intladon and employment chan its predecessor. This involves uncertainry and imperfect information wich the public altering the macroeconomic outcome consequent on a change in the monecary-regime. Moreover the Kydland-Prescott' analysis assumes a one-shot game, while in praaíce the central bank has a time-horizon induding several periods. This last sicuation refers to a repeaced game or a socalled supergame in which the strategic behavior of che player is dependent on the behavior of the other player in the past. This means thac in their aetual decision making all playecs have to take into account the future consequences of the current decisions. In this way during such a repeated game a periodic optimalisation is not possible because of the fact that in chosing the policy-scrategy the memory and che experiences of the players play an essential
role.
Backus and Driffill have shown the consequences with regard co the macroeconomic outcomes of a repeaced game (Backus and Driffill, 1985 a-b and Michener, 1989). These auchors assume a model with a"weak" and a"strong" policymaker. The reasoning goes as follows. The public does not know che rype of policymaker and has co gacher information about its idenciry. If the monetary authority is "weak" the macroeconomic outcomes in Table 1 will result. ln chac case the policymaker has always a scrong incenàve to organize an inflaàonary shock to obtain the. short-term benefits in terms of production and employmenc. However if the monecary authoriry is "strong" and independent, giving no room for disueàonary acàons, then a zero rate of inflation will always bc preferred. Private agencs know that the policymaker may be either of these rypes but they are ignoranc of which rype they are actually facing. Under such circumstances there is an incenàve for the "weak" policymaker to masquerade as a "strong" policymaker aiming at a downward adjusunent of inflaàonary expeccacions by the public.
policy-?~~
). ~. 5i~seN
credibility. This means that during a certain period the "weak" policymaker will be prepared to sacrifice the short-term benefits of a surprise intlation (investment in credibility) to build up a long-run reputaàon. In ocher words a"stock" of reputation will be build up, which can be used in the future if possibly the policymaker wanes to give prioriry to a sàmulation of economic activiry. [n these circumstances in their ex-post decisions policymakers will be faced with a socalled reputational trade-off (Persson, 1988). The higher the prioriry given to a building up of reputation, the greater the sacrifice in terms of a loss of real effeas in the short run and vice versa. The public is watching monetary authoriàes continuously and wants to find ouc whecher or not they stick to their initial anti-inflationary intentiuns. Economic agents try to unmask the policymaker to determine its true identiry.
During this learning-process the public will gradually revise the belief thac the policymaker is "strong" dependenc on the actual pace of int7acion. This degree of adjustment can be considered as a measure of poliey-credibiliry. [n this context credibiliry and the concomitanc build up of a reputacion can be understood as time-dependent state variables.
As long as monetary auchoricies are able to eliminate inflation and so stick to cheir initial intentions, the public will update the belief thac the policymaker is "strong" increasing its reputation. lf however at some time the policymaker appears to be "weak" by accepcing inflation and implemencing a discrecionary policy, the. public will punish this strategy-change immediately. This implies that the reputation will be lost direaly revealing the true idenciry of the policymaker. This view is also pointed out by the presidenc of the Ducch central bank, as he states, "?~ high race of inflation will be incorporated in the minds and expettations of the public and eliminaàng these expectations costs time. Therefore the elimination of an existenc inflacionary process is difficult and more costly than an avoidance of inflation" (Duisenberg, 1989, p. 19).
[f there is a final cime-horizon and full information the reputational-mechanism can not occur. For in a finite horízon game the policy maker will always raise intlation in the final period having no incentive to invest further in building up a repucaàon. However racional economic agents do understand this stracegy and will raise their inflacionary expectarions accordingly. Subsequently this reaction motivates the authorities to organize a surprise inflation in che penultimate period. But inflaàonary expectarions will be raised in this period as well preventing the arise of real effects. In this situation the game unravels backwards ulàmately resulting in a discrecionary or single stage Nash-equilibrium in all periods. This view comes from the theory of induscrial organization and is known as the socalled "chain store paradox" (Blackburn and Christensen, 1989). This implies that the working of the reputation-effect will only o~cur in a repeated game when tbere exists asytnmetric informaàon between the policymaker and che public.
Vlonetary Policy in a Game 245
scracegy at every point in the game, if ever it is called upon to do so" (Kreps and Wilson, 1982, p. 2~ï). Essennally ic deals with an inreraction process in a repeated game between the strategic behavior of policymakers and a d,vnamic Bayesian learning-process of the public. It is obvious thac such a learning-process will arise when in the course of time new policymakers come in. ln these circumstances the public is trying to find out and to learn the new policy-regime and co accounc for in its current decisions. During such a learning-period, indicaced by Taylor as a"transition co rational expeaacions", che meaning of reputatíon of policymakers comes abouc very sharply. In a very early stage a new policymaker can build up credibility and reputation by clearly revealing ics anti-inflacionary policy by organizing a restrictive monetary policy and even by accepting a contraetion of economic activiry.
The Backus-Driffill' analysis is restriaed in thac sense that che policymaker will be idencified bv the public eicher as a"weak" or as a"scrong" policymaker. Therefore when reputacion has disappeared it can not be build up again. This is a very scrong assumption. For if policymakers have revealed a bad behavior in the past it does noc mean that chis will always be the case in the future. ~fter the strong int7acionary developmenc in the sevencies monetary authonties in the United Staces as well as in ochtr industrialized countries have been able to change cheir policies and to reduce the aaual intlacion and the inflationarv expeccacions. In chis concext it is inceresting to nore thac the German central bank, which is a very independenc monetary authonry, has alwavs poinced ouc very clearlv ics preference for monecary stabiliry and has been successful in maintaining intlation and inflationary expectations on a relativel,v low leveL [n this wav the Bundesbank has generated credibiliry on internarional financial markecs and has build up the reputation of a"strong" policymaker and a stable monetan~ anchor in the financial world~).
The Reputational Equilibrium
In the novel licerature on game-cheory it is puc forward how building up a repucation through the implemencation of a pre-announced anti-inflacionary policy is a possibiliry co reduce rhe social costs (the loss oE employment) connected wich the elimination of inflation. In this concext Persson remarks, "Repucation may thus, fully or partly, substicute for formal precommicments, and lower or eliminate the costs of the credibiliry problem" (Persson, 1988, p. 522). This issue has been investigaced in a repeaced game by Barro and Gordon. In their analysis monetary auchorities will build up a reputation which to a cercain excent can generate the same credibiliry-effeas as an institutionally fixed monetary policy-rule. [n this way the inferior macroeconomic ouccomes of a discrecionary policy as described before can be improved or even avoided. The new issue in the Barro-Gordon analysis refers to the development of a reputacional-mechanism, which can prevent the extreme options of a formal precom-mitment of a zero inflacion-rule on the one hand and the discrerionary outcome (inflanon wichouc real effects) on che other hand. Their analysis does not deal wich an
') [n the game theoretic literature Vickers (1986) points out the meaning of a signalling-effect. This refers co the facz chac in the begirtning of the game che "scrong" policymaker will choose such a low rare of intlanon (even a recession) char the "weak" policymaker (a high-inElauon rype) will be decerred to masquerade him. This means rhac just from the beginning a clear signal will be given to the public abouc che true idennry of the policymaker faced with during the game. It also implies
246 J. J. SIJBEN
elaboration of the theoretical issues mentioned before, but with the introduction of the working of the reputacion-effect, which supports the realisaáon of a rule-equilibrium (zero-inflaáon) as well as muigates che incentive to cheat the public The Barro-Gordon contribution to che time-inconsiscency problem in monetary policy refers to the introduction of a reputational scrategy. This strategy makes it possible to regain panially the loss of credibiliry caused by discretionary aaions. This socalled ~rigger-mechanism describes how private sector inflationary expectations are revised in a rational way in response to the observed course of the actual inflationary process. In this way a connection will arise between the actual strategy-choice of policymakers with regard to employmenc and inflation on the one side and the concomitant inElationary expeaations of the public on che other side. [n building up a reputation central banks can avoid the inferior, sub-optimal, macroeconomic Nash-solution of a (consistenc) discretionary policy.
The potencial loss of repucation will enforce che policymaker in an informal way to stick to the pre-announced monerary poliey regime and secure a lower future average rate of inElation. The working of the reputational mechanism will mitigate the incentive to cheat che public, as was mentioned in the Kydland-Prescott analysis, and in any way will reduce the degree of non-cooperarive discretionary actions. Barro and Gordon clear(y point out that in case temptation (the incentive to organize a sucprise inflacion with its concomitanc shott-run real effects) is equal to the enforcement (the avoidance of a loss of reputation), the policymaker will stick to the pre-announced policy regime. However iE this is not the case priority will be given to cheating the public wich a concomitant loss of reputation and inferior macroeconomic outcomes in the future. In this way policymakers will be faced with quite a new trade-off, namel,v cheacing rhe public with real benefits in the short run versus a loss of reputation in the long run. For based on the game theoretic view cheating today implies higher inflanonary expeccations tomorrow.
Monetary Policy in a Game 247 in the next period by a loss of reputaáon. So the public can not be fooled systematically. This implies thac the policymaker has to pay for the shorc-term benefits of cheating in che previous period at a price equal to a loss of rcpucaáon in che fucure. Then in subsequent periods a non-cooperaáve game will aáse wich a sub-opcimal Nash-equilibrium. Barro and Gordon also explain thac in these circumstances the public behavior of scill sticking co a zero-inflaáon rule must be considered as irracional. Foc it is assumed that the public knows che policymaker's objective funcrion. Therefore che incenáve co cheac is always greaccr chan the pocenáal loss of repucacion, making ic impossible to enforce the ideal rule of zero-inflacion.
The monetary authority will maximize ia objeczive funcáon explicitly caking into accounc the racional behavior of the public. In considcring whecher to organize a surprise inflaáon today with ics concomitant short-run real effects or not the following trade-off results. The policymaker compares che current macroeconomic ouccome resulting from a surprise inflation of the public (:t ~ rt') with the outcome generated by the inflation rate which is in accordance wich che expected rate of inflacion ( n - n`). In this comparison attenáon is paid co che present valuoof the maaoeconomic costs in che next period which emerge from the faa that the cheated-public will adjust its inflacionary expectations upwards eliminaáng the initial real effects. As was poinced out before the public assumes that the policvmaker will continue its discretionary acáons (a 1 z' ). :~s long as the present value of these costs (deterrent-effect) are greater rhan the short-cerm benefits in cerms of an improvement of employmenc ( cemptacion-effect), the policymaker will change [he monetary regime and will choose a lower rate of inflacion chan is in accordance with a discrecionary policy.
?48 J. J. SIJBEN
the central bank. Barro and Gordon come to che conclusion that in these circumstances of a reputational-equilibrium the monecary authoriry announces a socalled "mid-value" of che rate of intlacion which on the one hand is lower than ics discretionary value and which on the other hand is positive. [n this concext chey remark, "Whether the ceputacional equilibrium will arise depends on the policymaker's weighing of the benefics from the two possible modes of behavior" (Barro and Gordon, 1983, p. 604-605 }.
The authors are also looking for an enforceable rule in a situacion charaaerized by asymmecric informacion, whereby the public does not know che prioriry and che discount rate of che policymaker. They also show rhac wich asymmetric information this rule is a weighted average of the ideal (zero-inflacion) rule and che inflacionary ouccome connected with a discrecionary policy. However in these circumstances che actual race of inflation changes with alterations in the prionry and in che discounc rate. This implies thac a higher prioricy given to employment and a higher race of discounc will be accompanied by a higher unexpec7ed race of inflacion. Conversely, policymakers will emphasize credibiliry and will invest in repucacion. In this context they poine out, "Conversely, the policymaker "bites the bullet" - that is, creates negacive inflation shocks" (Barro and Gordon, 1983 a, p. 118). In this way some scope will be built up for a fucure surprise intlacion when ac some cime higher prioriry may be given to a stimulation of economic activiry.
Taylor is criticizing rhis view and has some doubts abouc the meaning of the repucational-mechanism in a socalled positive theory of inflation. He remarks, "The fact that 1 suboprimallv high inftation rate is the only credible policy has been offered by Barro and Gordon as a reason why we have experienced high intlacion rates in recenr years. To some extent the incroduction of repucarion-effects inco che time inconsiscency model makes ic less attractive as a positive theory of inflationary políey: the larger are the reputation effects (for example che longer they last) the closer is the equilibrium inflation rate to zero" and further on, "In other well-recognized time inconsistency situacions, sociery seems to have found ways to institute che optimal (cooperative) policy" (Taylor, 1983, p. 125).
Subsequendy Canzoneri poincs ouc chac in che Barro-Gordon repucational equilib-rium the rate of inflacion will be constanc. According to him this is a rather odd conclusion because in practice it can be observed that in the course of cime the rate of intlation as well as che rate of monecary growth are altering substantially. He remarks, "The first weakness is that the Barro-Gordon solutton is, like Rogoff's too stable. If taken literally, it implies thac one should never expecr to see reversions to inflationary periods" (Canzoneri, 1985, p. 1064 and Rogoff, 1985). Moreover he states that the working of che reputational-mechanism is crucially dependent on the assumed punishmenc-strategy (loss of repucation) by the public. Therefore che monetary policy game wich an infinite horizon is charnaerized by several Nash-equilibria, buc there is no mechanism available explaining which choice will have to be made (Backus and Driffill, 1985).
Monecary Policy in a Game Z49
privare agents cannoc deduce immediacely which part of chese changes refers to a more scrucrural change in the policy regime and whích parc has to be ascribed co the imprecision with regard to controlling the money supply. In their analysis policy-credibility is no (áme dependenc) scate-variable, but will be determined by the speed with which privace agents realize rhat a shift in the polieymaker's preferences has caken place. This learning-process will be shortened and credibiliry will be higher the greater the precision of monecary concrol and vice versa.
In case of a precise monetary concrol some periods of monecary contraction will be sufficient to convince the public that the race of monetary growth wíll be reduced permanently and that the policy-regime has been changed. Therefore inflationary expectations will be revised downwards very soon. [n chis contexc the authors point out, "In this case a"cold turkey" disinflationary poliey is preferable to "gradualism" since a larger decrease in monetary expansion generates credibiliry relatively quick" (Cukierman and Meltzer, 1986, p. 1121). However if the precision of monetary control is very Iow, the public will be uncertain and unable to decide about a possible policy-shift based on the observed money supply figures. [n these circumscances the obtainment of credibiliry with regard to a disinflationary policy takes a long time and the social costs in terms of unemployment will be higher propornonally. Then a poliey of gradualism has co be preferred, enabling the public to adjust its expectacions gradually with lower social coscs. Subsequently che authors point ouc that under certain circumstances policymakers will be moávated to aim ar a cenain amount of ambiguity and secrecv in monetary poliey,.increasing asymmetric information. This means that policymakers may use the precision of monetary control as a policy inscrument to cheat the public and to regain some scope for scabilisacion policy.
~. Summarizing Conclusions
250 J. J. SIJBEN
regard to the inflation-unemployment trade-off. There always exisu an incenáve of polieymakers to cheat the public by organizing an inflaáonary-shock, aimed at obcaining short-run benencs in terms of an improvemenc of production and employ-ment. [f the ultimate ouccome of monetary policy is also dependenc on public behavior, the policymaker's abiliry to change its original plan has two consequences. Pnmarily if che public knows that the policymaker might change its original incentions (incentive to cheac), it will act in the decision-making process as if rhe policymaker is actually implementing the policy shift. Then this informacion will face the policymaker wich a difficult choice at the moment che new policy has co be implemenced. Eicher the original plan will be changed, given the new economic environmenc, (time-inconsiscency) or the policymaker is giving prioriry to the long-run benefics in terms of credibiliry and reputation and will scick to the original opámal plan. Therefore the opcimal poliey mighc become rime-inconsistenc, resulting in a loss of credibiliry and repucaáon. In section 3 the asymmecric informacion in the game cheoretic Eramework is emphasized, dealing wich che crucial concepts of credibiliry and reputation of the policymaker.
Due to the experiences wich severe inflacions in the seventies, recenc economic research on che rules versus discretion debare is strongly emphasizing the role of policy-credibiliry and reputaáon of central banks to prevent a rekindling of inflacion. This development has revitalized the old concroversy abouc rules in monetary policy versus discrecionary monecary aaions. The "time-inconsistency" problem is re-emphasizing very clearly thac che concepts of uncertainry, informacion and expecta-tions play a significanc role in the inceraaion process between the polieymaker and che public and therefore are very crucial for the ultimace course of the economic process. The empirical research in the field of credibiliry and reputation wich regard to monecary policy is still in its infaney and many problems have to be resolved. (Baxtec, 1985, Blanchard, 1984). The hypochesis of "time inconsisteney" has also to be tested empirically. This might be realized by regarding che evidence of a systematic "inflation bias" in those countries where the central bank is less independenc and more subservient to the government.
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Summary
The purpose of this paper is co give a non-technical review oE the currenc issues in the
international lirerature with regard to che game-theoreác approach in monecary policy. Aher the ~Lucas-Sargent-Wallace ineffectiveness hypothesis in the sevencies a new policy-issue came abou[ dealing with the nme-inconsistency of policy makers. This quesáon is closely rela[ed [o ché concepts of policy-credibiliry and reputation of ceneral banks. According co the game-cheoreác
approach the interarnon process between the s[racegic behaviour of the policymaker and the
Monetary Policy in a Game 253
poGcy has to bc abandoned, because the long-run outcomes in terms of inflation and employmenc will bc less benefiaal (Nash-eqwlibnum! rhan in [he case of a precommirment in mone[ary policy (rule).
Zusammen~assung
Dieser ~lrrikel versucht eine nich[-technische Ubcrsicht uber in dcr íncernarionalen Lirerarur diskuticrte Anwendungen der Spielthcorie auf geldpolinsche Fragestellungen zu gcben (Debatte uber regelgebundene versus diskretionàre Politik). Nach dcr Ineffektivitàtshypochese von Lucas, Sargcnt und Wallace in den siebziger Jahren trat mi[ der Zcicinkonsisrenz der Polirik ein neues Politikproblem in den Vordcrgrund. Diescs Problem ist eng mit den Konzep[en der Glaubwur-digkcic und Repuration von Zencralbanken verbunden. Der spielcheorecischen Analyse zufolge fiíhrt die Inreration zwischen dcn srrategisch handelnden politischen tlktoren und der racional agierenden Óffendichkeit zu eincr cndogen dccermimerten Gcldpolitik. Dies impliziert, daó man diskredonàre Politik aufgeben mud, weil ihre langfrisrigen Resultace bezuglich [ntla[ion und Bescháhigung schlechter wáren (Nash-G(eichgewich[) als díe riner er ~nce angekiindig[en Gledmengenregel.
Reprint Series, CentER, Tilburg Univetsity, The Netherlands:
No. t
No. 2
No. 3
G. Marini and F. van der Ploeg, Monetary and fiscal policy in an optimising model with capital ac:cumulation and finite lives, The Economic Joumal, vol. 98, no. 392, 1988, pp. 772 - 786.
F. van der Ploeg, International policy coordination in interdependent monetary economies, Journa! oj lntemational Economics, vol. 25, 1988, pp. 1- 23. A.P. Barten, The history of Dutch macroeconomic modelling (1936-1986), in W. Driehuis, M.M.G. Fase and H. den Hartog (eds.), Challenges jor Macroeconomic
Modelling, Contributions to Economic Analysis 178, Amsterdam: North-Holland,
1988, PP. 39 - 88.
No. 4 F. van der Ploeg, Disposable income, unemployment, intlation and state spending
in a dynamic political-economic model, Public Choice, vol. 60, 1989, pp. 211 - 239.
No. 5 Th. ten Raa and F. van der Ploeg, A statistical approach to the problem of negatives in input-output analysis, Economic Modelling, vol. 6, no. 1, 1989, pp. 2 - 19.
No. 6 E. van Damme, Renegotia[ion-proof equilibria in repeated prisoners' dilemma,
Joumal of Economic Theory, vol. 47, no. 1, 1989, pp. 206 - 217.
No. 7 C. Mulder and F. van der Ploeg, Trade unions, investment and employment in
a small open economy: a Dutch perspective, in J. Muysken and C. de Neubourg (eds.), Unemploymentin Europe, London: The Macmillan Press Ltd, 1989, pp. 200
- 229.
No. 8 Th. van de Klundert and F. van der Ploeg, Wage rigidity and capital mobility in an optimizing model of a small open economy, De Economut, vol. 137, nr. 1,
1989, pp. 47 - 75.
No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergen model revisited, Economic Modelling, vol. 6, no. 2, 1989, pp. 203 - 219.
No. 10 F. van der Ploeg and A.J. de Zeeuw, Conflict over arms accumulation in market and command economies, in F. van der Ploeg and A.J. de Zeeuw (eds.), Dvnamic
Policy Gumes in Economics, Contributions to Economic Analysis 181,
Amster-dam: Elsevier Science PubGshers B.V. (North-Holland), I989, pp. 91 - 119. No. 11 1. Driffill, Macroeconomic policy games with incomplete information: some
extensions, in F. van der Ploeg and A.J. de Zeeuw ( eds.), Dvnamic Policy Cames
in Economics, Contributions to Economic Analysis 181, Amsterdam: Elsevier
Science Publishers B.V. (North-Holland), 1989, pp. 289 - 322.
No. 12 F. van der Ploeg, Towards monetary integration in Europe, in P. De Grauwe et aL, De Europese Monetaire Integratie: vier visies, Wetenschappelijke Raad voor het
No. 13 R.J.M. Alessie and A. Kapteyn, Consumption, savings and demographv, in A. Wenig, K.F. Zimmermann (eds.), Demographic CfuinSe and Econoinic Development, Berlin~Heidelberg: Springer-Verlag, 1989, pp. Z7~ -;0~.
No. 14 A. Hoque, J.R. Magnus and B. Pesaran, The exact multi-period mean-square forecast error for the first-order autoregressive model, Joumal of Econnmerncs, vol. 39, no. 3, 1988, pp. 327 - 346.
No. 15 R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidity constraints on eonsump[ion: estima[ion from household panel data, European Economic Review, vol. 33, no. 2~3, 1989, pp. 547 - 555.
No. l6 A. Holly and J.R. Magnus, A note on instrumental variables and maximum
likeli-hood estimation procedures, Annales d'Économie et de S[atistulue, no. 10, April-June, 1988, pp. 121 - 138.
No. l7 P. ten Hacken, A. Kapteyn and I. Woittiez, Unemployment benefits and the labor market, a micro~macro approach, in B.A. Gustafsson and N. Anders Klevmarken (eds.), The Political Economy of Social Securiry, Contributions to Economic Analysis 179, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1989, pp. l43 - 164.
No. 18 T. Wansbeek and A. Kapteyn, Estimation of the error-components model with incomplete panels, Journal of Economerncs, vol. 41, no. 3, 1989, pp. 341 - 361.
No. ]9 A. Kapteyn, P. Kooreman and R. Willemse, Some methodological issues in the
implementation of subjective poverty definitions, The Journal of Human
Resources, vol. 23, no. 2, 1988, pp. 222 - 242.
No. 20 Th. van de Klundert and F. van der Plceg, Fiscal policy and finite lives in interdependent economies with real and nominal wage rigidity, Ozford Economic
Papers, vol. 41, no. 3, 1989, pp. 459 - 489.
No. 21 J.R. Magnus and B. Pesaran, The exaM multi-period mean-square forecast error for the fu-st-order autoregressive model with an intercept, Joumal oj
Economerricr, vol. 42, no. 2, 1989, pp. 157 - 179.
No. 22 F. van der Plceg, Two essays on political economy: (i) The political economy of overvaluation, The Economic Joumal, vol. 99, no. 397, 1989, pp. 850 - 855; (ii) Election outcomes and the stockmarket, European Jounwl of Political Economy, vol. 5, no. 1, 1989, pp. 21 - 30.
No. 23 J.R. Magnus and A.D. Woodland, On the mazimum likelihood estimation of
multivariate regression models containing serially correlated error components,
Intematéonal Economic Review, vol. 29, no. 4, 1988, PP. 707 - 725.
No. 24 A.J.J. Talman and Y. Yamamoto, A simplicial algorithm for stationary point problems on polytopes, Mathematics of Operations Research, vol. l4, no. 3, 1989, pp. 383 - 399.
No. 25 E. van Damme, Stable equílibria and forward induction, Iournal of Economic
